Problem: Let $f(x)=-2\log_7(x)$. Find $f'(x)$. Choose 1 answer: Choose 1 answer: (Choice A) A $-\dfrac{2}{\log_7(x)}$ (Choice B) B $-\dfrac{2}{x\log_7(x)}$ (Choice C) C $-\dfrac{2}{7\ln(x)}$ (Choice D) D $-\dfrac{2}{x\ln(7)}$
Answer: The expression for $f(x)$ includes a logarithmic term. Remember that the derivative of the general logarithmic term $\log_a(x)$ (where $a$ is any positive constant and $a\neq 1$ ) is $\dfrac{1}{\ln(a)\cdot x}$. Put another way, $\dfrac{d}{dx}[\log_a(x)]=\dfrac{1}{\ln(a)\cdot x}$. [Is there an easy way to memorize that?] We can use this to find the derivative of the function as shown below. $\begin{aligned} f'(x)&=\dfrac{d}{dx}[-2\log_7(x)] \\\\ &=-2\dfrac{d}{dx}[\log_7(x)] \\\\ &=-2\cdot\dfrac{1}{\ln(7)x} \\\\ &=-\dfrac{2}{x\ln(7)} \end{aligned}$ In conclusion, $f'(x)=-\dfrac{2}{x\ln(7)}$.